21.04.24 - Osr - Star Co-Sc - Jee-Adv - 2023 - P1 - Gta-7 (P2) - QP

Sec: OSR.
IIT_*CO-SC Date: 21-04-24

Time: 3HRS Max. Marks: 180
Name of the Student: ___________________ H.T. NO:
21-04-24_OSR.STAR CO-SUPER CHAINA_JEE-ADV_GTA-7(P2)_SYLLABUS
MATHEMATICS: TOTAL SYLLABUS
PHYSICS: TOTAL SYLLABUS
CHEMISTRY: TOTAL SYLLABUS

Narayana IIT Academy 21-04-24_OSR.IIT_*CO-SC_JEE-Adv_GTA-7(P2)_Q’P
TIME:3hrs IMPORTANT INSTRUCTIONS Max Marks: 180
MATHEMATICS
+Ve - Ve No.of Total
Section Question Type
Marks Marks Qs marks
Sec – I (Q.N : 1 – 4) Questions with Single Correct Options +3 -2 4 12
Questions with Multiple Correct Choice
Sec – II (Q.N : 5 – 7) +4 -1 3 12
(partial marking scheme) (+1,0)
Sec – III (Q.N : 8 – 13) Questions with Non-Negative Integer type +4 0 6 24
Questions with Comprehension Type With
Sec – IV (Q.N : 14 – 17) Numerical value type +3 -1 4 12
(3 Comprehensions – 2 + 2 = 4Q)
Total 17 60
PHYSICS
Sec – I (Q.N : 18 – 21) +3 -2 4 12
Sec – II (Q.N : 22 – 24) Questions with Single Correct Options +4 -1 3 12
Sec – IV (Q.N : 31 – 34) Questions with MATRIX MATCH +3 -1 4 12
Total 17 60
CHEMISTRY
Sec – I (Q.N : 35 – 38) +3 -2 4 12
Sec – II (Q.N : 39 – 41) Questions with Single Correct Options +4 -1 3 12
Sec – IV (Q.N : 48 –51) Questions with MATRIX MATCH +3 -1 4 12
Total 17 60
OSR.IIT_*CO-SC Page. No. 2

MATHEMATICS MAX.MARKS: 60
SECTION–I (Maximum Marks: 12)

This section contains FOUR (04) questions.
Each question has FOUR options for correct answer(s). ONLY ONE of these four option is the correct answer.
For each question, choose the correct option corresponding to the correct answer.
Answer to each question will be evaluated according to the following marking scheme:
Full Marks : +3 If only the correct option is chosen.
Zero Marks: 0 If none of the option is chosen.(i.e the question is un answered)
Negative Marks: -1 In all other cases.
 5a b  T
1. Let A    be a 2  2 matrix such that A. adj(A)  A.A . Then
3 2
A) a unique such A exists B) Two distinct such A exist
C) No such A exists D) Infinitely many such A exist

n 20
 1 
2. Let f (n)    cot 1  k   tan 1 k  , k  0 . If the value of  (f (n)  f (n  1)) is m , then the
k  n n 2
value of m is
A) 100 B) 98 C) 99 D) 117
x  2 y 1 z  2
3. If the line   lies in the plane x  3y  z    0 , then the number of
3 5 2
integral solutions of the equation x1x 2 x 3  2 is equal to
A) 27 B) 54 C) 108 D) 216
3  cos(x)
4. Let x, y  R such that cos(xy)  cos (xy  x)  . Consider the following statements:
2
(I) x must be an even multiple of 
(II) xy must be an even multiple of 
(III) y must be rational
(IV) x must be irrational
How many of the above statements are true
A) 1 B) 2 C) 3 D) 4

SECTION - II (Maximum Marks : 12)
This section contains SIX (03) questions.
Each question has FOUR options for correct answer(s). ONE OR MORE THAN ONE of these four option(s) is
(are) correct option(s).
For each question, choose the correct option(s) to answer the question.
Full Marks : +4 If only (all) the correct option(s) is (are) chosen.
Partial Marks: +3 If all the four options are correct but ONLY three options are chosen.
Partial Marks: +2 If three or more options are correct but ONLY two options are chosen, both of which are
correct options.
Partial Marks : +1 If two or more options are correct but ONLY one option is chosen and it is a correct
option.
Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered).
(x  y)(1  xy)
5. Let f (x, y)  , where x, y  R . Which of the following values can be assumed
(1  x 2 )(1  y 2 )
by f?
1 2 3
A) 0 B) C)  D)
2 5 4
6. Let A be a 2  2 square matrix with real entries such that det(A 2  4I2 )  0 . Then, which of
the following is (are) necessarily true? ( I 2 is the identity matrix of order 2)
A) A 2  4I2 is a null matrix B) A 2  A is a null matrix
C) det (A 2  A  4I2 )  4 D) det (A 2  A  5I2 )  5

1
7. Let f : R  (0, ) is a continuous function such that f (x)   e x  e  x x  R . Then, area
f (x)
bounded by y  f (x) , x – axis, ordinate x  1 and ordinate x  1 can be equal to
1 1 2
A) e  B) 2e  2 C) 2  D) 2 
e e e
SECTION-III (Maximum Marks : 24)

This section contains THREE (06) questions. The answer to each question is a NON-NEGATIVE INTEGER
For each question, enter the correct integer corresponding to the answer using the mouse and the on-screen
virtual numeric keypad in the place designated to enter answer.
Full Marks: +4 If ONLY the correct numerical value is entered as answer.
Zero Marks: 0 In all other cases.
8. Let x, y, z  Z . The number of ordered triplets (x, y, z) such that
| x  1|  | y  2 |  | z  3 |  | x  4 |  | y  7 | 2 | z  10 |  21 is equal to

2
tan x dx
9. If I   2 2
, then the value of [2024I] is equal to (where [x] is the GIF)
0 4 ln (tan x)  

e
 I 
10. Let I n   (log e x)n dx, n  I, n  0 . The value of  7  is equal to (where [x] is the GIF)
1  I5  I 6 
11. The differential equation of the family of tangent lines of the curve y 2  x 3 is given by
3
 dy   dy 
   x   y  0
 dx   dx 
m
where  is a constant. If   , where m, n are coprime positive integers, then the value
n
of m  n 

k 1
12. Let S   4 2
. The value of is equal to
k 1 16k  24k  25 S
 (n 100)2 1 
13. The value of lim    is equal to
n   r 
 r (n 10) 2

SECTION - IV (Maximum Marks : 12)

This section contains TWO (2) Paragraphs. Based on each paragraph, there are 2 questions.
The answer to each question is a NUMERICAL VALUE.
For each question, enter the correct numerical value of the answer using the mouse and the on-screen virtual
numeric keypad in the place designated to enter answer. If the numerical value has more than two decimal
places truncate/round- off the value to TWO decimal places.
Paragraph-I for Question Nos. 14 and 15:
p2
If  (p1r  3).(p 2 Cr )  (210 )(38) where p1 , p 2  N then
r 0
14. p1 is equal to
15. p 2 is equal to
Paragraph-II for Question Nos. 16 and 17:

x4  x2 p
Let f (x)  6 3
; x  1 . Let the absolute maximum value of f(x) be equal to , where
x  2x  1 q
a b
p and q are co-prime natural numbers. Also, this maximum occurs at x  where a,
c
b and c are pairwise co-prime natural numbers. Let p1  p  q  1 and p 2  abc . Then,
16. p1 is equal to
17. p 2 is equal to

PHYSICS MAX.MARKS: 60

18. A Lightweight thin wire ring is hanging on a soap film that is held in a circular frame.
The mass of the ring is m, its radius R, the surface tension of the film is  , the diameter
of the frame D > 2R. The frame and ring are horizontal, their centres are on the same
vertical line. What is the distance from the plane of the ring to the plane of the frame?
Mass of the film can be neglected in comparison with the mass of the ring, Also
mg  R .
mg D mg  D 
A) h  n B) h   R
4 2R 4R  2 
mg  D  mg  D 
C) h   R D) h  tan 1  
2R  2  4  2R 
19. A parallel plate capacitor with a capacity C with an air dielectric consists of two large
plates, located very close to each other. One of plates is uncharged, the other plate has a
charge Q. We connect the plates with a conductor having a high resistance R. Estimate
the amount of heat that will be dissipated in the conductor.
Q2 Q2 Q2 Q2
A) B) C) D)
2C 4C 6C 8C
20. In a YDSE apparatus, the intensity at the central maxima is 4 W / m 2 . Both the slits are of
equal width and the wavelength of the light used is 600 nm. An extremely thin glass
plate of thickness 200 nm and refractive index 1.5 is placed in front of one of the slits
and it is seen that the intensity at the centre of the screen (where earlier there was central
maxima) is now 1.75 W / m 2 . It is suspected that the glass plate absorbs some light. What
is the new intensity at the central maxima?
A) 3.00 W / m 2 B) 2.25 W / m 2 C) 3.60 W / m 2 D) 2.40 W / m 2

21. Atoms of a hydrogen like gas are in a particular excited energy level. When these atoms
de-excite, they emit photons of different energies. Maximum and minimum energies of
emitted photons are E max  52.224 eV and E min  1.224 eV respectively. Calculate principal
quantum number of initially excited energy level. (Ionisation energy of hydrogen atom is
13.6 eV)
A) 6 B) 4 C) 5 D) 3

correct options.
option.
22. A wire 30.0 cm long is held parallel to and 80.0 cm above a long wire carrying 200 A
and resting on the floor (Fig.) The 30.0 cm wire is released at t = 0 and falls, remaining
parallel with the current – carrying wire as it falls. Assume that the falling wire
accelerates at 10m / s 2 (The wire is falling freely).
A) The left end of the wire will be at a higher potential as it falls.
B) The emf is maximum when it is about to hit the wire
C) The emf at t = 0.2 sec is 0.04 mV
D) The emf is maximum at the initial moment.

23. Consider an ohmmeter used to measure the resistance R connected to it. It is basically an
ammeter whose scale is calibrated to measure the resistance R. The internal resistance of
the battery is r, its emf is E, and the internal resistance of the ammeter is r1 . The least
count of the ammeter is I .
A) The error in the measurement of R is minimum when current is maximum

B) The error in the measurement of R is maximum when current maximum
C) The ohmmeter scale is linear (the divisions are equally spaced)
D) The sensitivity of the ohmmeter   is maximum when current is maximum

dI

dR 
24. They say that in Snell’s archive they found an optical scheme, which depicted lens, Real
object and its real image. The object and the image are of the same size and shape, and
the longer line of the object lies on the optical axis of the lens. The image and object
both are shown here. Focal length of the lens = f. Based on the given diagram, select the
correct options.
A) O should be at a distance of 2f from the lens

B) The lens should be on right – hand side of the object
C) The focal length of the lens is 2a where a is length of one square grid
D) The lens is converging in nature

25. A mercury thermometer reads 80ºC when the mercury is at 5.2 cm mark and 60ºC when
the mercury is at 3.9 cm mark. If the temperature when the mercury level is at 2.6 cm
mark is T (in degree Celsius), find T – 35
26. A standing wave with a frequency of 1000 Hz in a column of mixture of unknown gases
CP
at 27ºC produces nodes that are 0.1 m apart. Find the value of r  for the mixture.
CV
Use R  8.3J / mol  K and molar mass of mixture 124.5 gm/mol. CP & CV are the molar
heat capacities at constant pressure and constant volume respectively.
L
27. Two rigid rods of same material and of length L and are placed in equilibrium on a
2
smooth horizontal plane at temperature T as shown in figure. Springs are in their natural
length. If the temperature is increased by T amount energy stored in springs is
9
k 2 L2 T 2 , where k is the spring constant of spring (A) and  is the coefficient of
5
linear expansion of material. (Note : Neglect the change in spring constant due to
increase in temperature) Ignore thermal expansion of springs.
28. The main scale of a vernier callipers reads in millimeter and its vernier is divided into
10 divisions which coincides with 9 divisions of the main scale. The reading for shown
23x
situation is found to be mm . Find the value of x .
10

29. A massless scalene triangular platform (sides a, b and c) has a point mass m at its
centroid. It is suspended from ceiling by three light springs of constants k, k and 2k
respectively as show in the figure. In equilibrium the platform is perfectly horizontal.
Now from this configuration, a small vertical impulse is imparted to m such that it
acquires a small vertical velocity u. Calculate:
(A) Time Period of Oscillation
(B) Amplitude of oscillation of m
am
The time period of oscillation of the system is given as T  2 where a and b are
bk
coprime
Natural numbers report the sum of digits of the number (a + b).
30. A non – conducting cylindrical vessel of length 3 is placed horizontally and is divided
into three parts by two easily moving pistons having low thermal conductivity. As
shown three parts contain H 2 , He and CO2 gas at initial temperatures 1  372º C,
2  15º C, 3  157º C respectively. The initial length and pressure of each part are  and
n
P0 respectively. If the length of middle part is finally , Then report the sum of digits
30
of the number n.

Recently there has been a lot of hue and cry over the possibility of an asteroid hitting the
earth. NASA reports that it has spotted an asteroid Bennu which may possible hit the
earth somewhere in 22nd century. The probability of its hitting the earth’s surface is only

one in 2000, yet social media sites are expert in spreading rumour about its possible
impact. Assume that it was at rest at a large distance from earth and it is moving only
under the influence of earth’s gravitational field. Neglect the influence of any other
celestial body like sun, moon etc. on the asteroid as well as the earth. The size of Bennu
is much smaller than earth. (Take : escape speed from earth’s surface as 11 km/s).If the
asteroid were to hit the surface of the earth normally at the equator and gets embedded in
the earth just below the surface, the duration of the day would become 30 hrs.
31. What is the ratio mass of the asteroid / mass of earth?
32. What is the velocity v (in km/s) of asteroid just before hitting the earth? Fill the value of
v2 .
Consider a small disc of mass m and charge + Q on the surface of a rough inclined
plane. A uniform magnetic field exists perpendicular to the inclined plane as shown. The
inclined plane is inclined at an angle of  to the horizontal. The inclined plane is very
long and wide. The coefficient of friction is  .
Take   60º , m  80gm, q  100mC, B  0.4T .
(tan )
33. If   , assuming that disc attains a certain terminal velocity after sufficiently long
2
time, what is the speed of the disc (in m/s) after a long time? The disc is released from
rest.
34. Now assume that the width of the inclined plane is 2L. At the initial moment, the disc is
at a distance of L from the left edge. Also assume that the friction coefficient is   tan  .
What can be the minimum initial speed along the inclined plane (in cm/s) so that the disc
leaves the inclined plane from one of the the two side edges ? Take : L = 0.3 m
(Assume that particle’s height above the ground decreases monotonically until leaving
one of the side edges)

CHEMISTRY MAX.MARKS: 60

35.
A) B)
C) D)
36. Complete the equations:
Final product during given reaction is:
A) B) C) D)

37. “Roasting of ore followed by reduction with coke” is the common method of the
extraction of
A) Zinc from Calamine B) Zinc from Sphalerite
C) Silver from Argentite D) Calcium from Dolomite
38. Which of the following statements is incorrect for a closed system of constant
composition, and in the absence of any additional (non – expansion) work,
A) For a reversible change, the sum of dw and dq is equal to the sum of TdS and
– pdV
B) For an irreversible change, TdS > dq
C) For an irreversible change, pdV  dw
D) For an irreversible change, the sum of dw and dq is equal to the sum of TdS and
 pdV

correct options.
option.
39.
X and Y respectively can be:

A)
B)
C)
D)
40. Cl2  dry slaked lime  Product
Select CORRECT about above reaction & product of reaction.
A) It is disproportionation reaction
B) Product gives brick red colour on flame
C) Product can decolorize coloured organic matter
D) Oxidation number of chlorine is 1 & 5 in product
41. ‘X’ and ‘Y’ are two elements which form X 2 Y3 and XY2 molecules. If 0.15 mole of
X 2 Y3 and XY2 weighs 15.9 g m & 9.3 gm respectively. If oxidation state of ‘Y’ in both
compounds is +2, then correct options are:
A) Atomic weight of X = 26 & Y = 18
B) Equivalent weight of X 2 Y3  17.6 & XY2  15.5
C) Atomic weight of X = 26 & Y = 36
D) Equivalent weight of X 2 Y3  26.66 & XY2  24.5

42. How many enantiomeric pairs are formed on monochlorination of 3 – methylpentane?
43. Find sum of the oxidation number of sulphur containing product in all the given
reactions

H /
i. Mn 2  S2O82  

ii. Cr2O72  SO32 
H

iii. MnO 4  S2O32 
OH


iv. MnO 4  H 2S 
H

v. MnO 4  SO2 
H
in presence
44. Auriferrous rock  NaCN of air
  "X"
Where “X” is the complex made by Au. Find the ratio of  - bonds to  - bonds in “X”
(without considering the synergic bonding)
45. Nitrogen gas adsorbed on charcoal to the extent of 0.387 cm3 / g at a pressure of 1.6 atm
and at temperature of 200K, but at 250 K the same amount of adsorption was achieved
only when the pressure was increased to 32 atm. The magnitude of molar enthalpy of
adsorption (in kcal/mol) of nitrogen on charcoal is (ln 20 = 3.0)
46. The electron of a hydrogen atom is in its nth Bohr orbit has de Broglie wavelength
13.4Aº. The value of n is ________________(rounded up to the nearest integer)
47. The total number of compounds having at least one bridging oxo group among the
molecules given below is ____.
N 2 O 4 , N 2 O5 , P4 O6 , S3 O9 , H 4 P2 O5 , H 5 P3 O10 , H 2S2O7 , H 2S2 O5

O
Excess
(P)
of HI
O
(R)
48. Find the number of moles of HI required to react 1 mole of R completely.
49. The number of Iodine Atom present in a molecule of P?
So ln Conductivity
0.01 M HA so ln (weak acid) 3.8 105 S cm1
100cm3 of 0.01M HA so ln  1 mmol
of NaOH 60  105 Scm 1
(neglect vol. change)
100cm3 of 0.04M NaA so ln 160 105 Scm 1
Also, m , Na   50S cm2 mol1
 m , H   350 S cm 2 mol1
50. Determine  m , A  in S cm 2 mol1
51. Determine percentage dissociation of HA at 0.01 M

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Sec: OSR.

IIT_*CO-SC Date: 21-04-24

Name of the Student: ___________________ H.T. NO:

21-04-24_OSR.STAR CO-SUPER CHAINA_JEE-ADV_GTA-7(P2)_SYLLABUS

MATHEMATICS: TOTAL SYLLABUS

PHYSICS: TOTAL SYLLABUS

CHEMISTRY: TOTAL SYLLABUS

OSR.IIT_*CO-SC Page. No. 2

SECTION–I (Maximum Marks: 12)

A) a unique such A exists B) Two distinct such A exist

C) No such A exists D) Infinitely many such A exist

(I) x must be an even multiple of 

(II) xy must be an even multiple of 

(III) y must be rational

(IV) x must be irrational

How many of the above statements are true

OSR.IIT_*CO-SC Page. No. 3

A) A 2  4I2 is a null matrix B) A 2  A is a null matrix

C) det (A 2  A  4I2 )  4 D) det (A 2  A  5I2 )  5

SECTION-III (Maximum Marks : 24)

OSR.IIT_*CO-SC Page. No. 4

SECTION - IV (Maximum Marks : 12)

Paragraph-II for Question Nos. 16 and 17:

OSR.IIT_*CO-SC Page. No. 5

SECTION–I (Maximum Marks: 12)

OSR.IIT_*CO-SC Page. No. 6

SECTION - II (Maximum Marks : 12)

A) The left end of the wire will be at a higher potential as it falls.

B) The emf is maximum when it is about to hit the wire

C) The emf at t = 0.2 sec is 0.04 mV

D) The emf is maximum at the initial moment.

OSR.IIT_*CO-SC Page. No. 7

A) The error in the measurement of R is minimum when current is maximum

D) The sensitivity of the ohmmeter   is maximum when current is maximum

A) O should be at a distance of 2f from the lens

OSR.IIT_*CO-SC Page. No. 8

OSR.IIT_*CO-SC Page. No. 9

SECTION - IV (Maximum Marks : 12)

OSR.IIT_*CO-SC Page. No. 10

OSR.IIT_*CO-SC Page. No. 11

SECTION–I (Maximum Marks: 12)

36. Complete the equations:

Final product during given reaction is:

OSR.IIT_*CO-SC Page. No. 12

A) Zinc from Calamine B) Zinc from Sphalerite

C) Silver from Argentite D) Calcium from Dolomite

B) For an irreversible change, TdS > dq

C) For an irreversible change, pdV  dw

SECTION - II (Maximum Marks : 12)

X and Y respectively can be:

OSR.IIT_*CO-SC Page. No. 13

Select CORRECT about above reaction & product of reaction.

B) Product gives brick red colour on flame

C) Product can decolorize coloured organic matter

D) Oxidation number of chlorine is 1 & 5 in product

OSR.IIT_*CO-SC Page. No. 14

N 2 O 4 , N 2 O5 , P4 O6 , S3 O9 , H 4 P2 O5 , H 5 P3 O10 , H 2S2O7 , H 2S2 O5

OSR.IIT_*CO-SC Page. No. 15

48. Find the number of moles of HI required to react 1 mole of R completely.